On the expansions of real numbers in two integer bases

Yann Bugeaud, Dong Han Kim

Published 2015-12-22Version 1

Let $r \ge 2$ and $s \ge 2$ be distinct integers. We establish that, if $r$ and $s$ are multiplicatively independent, then the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words. We also discuss the case where $r$ and $s$ are multiplicatively dependent.